Reconfigurable Real-Time Middleware for Distributed Cyber-Physical Systems with Aperiodic Events
An Introduction to Signals & Systems · 2014. 4. 1. · Continuous-time and discrete-time signals...
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1 An Introduction to Signals & Systems © 2014 School of Information Technology and Electrical Engineering at The University of Queensland http://elec3004.com Lecture Schedule: 2 6 March 2014 ELEC 3004: Systems Week Date Lecture Title 1 4-Mar Introduction & Systems Overview 6-Mar [Linear Dynamical Systems] 2 11-Mar Signals as Vectors & Systems as Maps 13-Mar [Signals] 3 18-Mar Sampling & Data Acquisition & Antialiasing Filters 20-Mar [Discrete Signals] 4 25-Mar Filter Analysis & Filter Design 27-Mar [Filters] 5 1-Apr Digital Filters 3-Apr [Digital Filters] 6 8-Apr Discrete Systems & Z-Transforms 10-Apr [Z-Transforms] 7 15-Apr Convolution & FT & DFT 17-Apr Frequency Response 8 29-Apr Introduction to Control 1-May [Feedback] 9 6-May Introduction to Digital Control 8-May [Digitial Control] 10 13-May Stability of Digital Systems 15-May [Stability] 11 20-May State-Space 22-May Controllability & Observability 12 27-May PID Control & System Identification 29-May Digitial Control System Hardware 13 3-Jun Applications in Industry & Information Theory & Communications 5-Jun Summary and Course Review
Transcript of An Introduction to Signals & Systems · 2014. 4. 1. · Continuous-time and discrete-time signals...
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An Introduction to Signals & Systems
© 2014 School of Information Technology and Electrical Engineering at The University of Queensland
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http://elec3004.com
Lecture Schedule:
2 6 March 2014 ELEC 3004: Systems
Week Date Lecture Title
1 4-Mar Introduction & Systems Overview
6-Mar [Linear Dynamical Systems]
2 11-Mar Signals as Vectors & Systems as Maps
13-Mar [Signals]
3 18-Mar Sampling & Data Acquisition & Antialiasing Filters
20-Mar [Discrete Signals]
4 25-Mar Filter Analysis & Filter Design
27-Mar [Filters]
5 1-Apr Digital Filters
3-Apr [Digital Filters]
6 8-Apr Discrete Systems & Z-Transforms
10-Apr [Z-Transforms]
7 15-Apr Convolution & FT & DFT
17-Apr Frequency Response
8 29-Apr Introduction to Control
1-May [Feedback]
9 6-May Introduction to Digital Control
8-May [Digitial Control]
10 13-May Stability of Digital Systems
15-May [Stability]
11 20-May State-Space
22-May Controllability & Observability
12 27-May PID Control & System Identification
29-May Digitial Control System Hardware
13 3-Jun Applications in Industry & Information Theory & Communications
5-Jun Summary and Course Review
http://itee.uq.edu.au/~metr4202/http://itee.uq.edu.au/~metr4202/http://creativecommons.org/licenses/by-nc-sa/3.0/au/deed.en_UShttp://elec3004.com/http://elec3004.com/
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Signals
• Communicates information
• Varies in Time and/or Space
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What is a Signal?
Everywhere:
• The type
• The screen
• Your 5 senses
• Telephony
• Interest rates
• Leaf colour
Temperature
Distance
Force
Speed
F(x)
Signal
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Understanding & Classifying Signals
Metrics:
• Size
• Signal Energy
• Signal Power
• Frequency
• Phase
• Entropy
– Deterministic signals
– Random signals
Classifications:
• Continuous-Time
• Discrete-Time
• Analog
• Digital
• Even
• Odd
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• The size of any entity is a number that indicates the largeness
or strength of that entity. Generally speaking, the signal
amplitude varies with time.
• However, this will be a defective measure because even for a
large signal x(t), its positive and negative areas could cancel
each other, indicating a signal of small size.
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Signal Size
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• Consider the area under a signal x(t) as a possible measure of
its size, because it takes account not only of the amplitude but
also of the duration.
• Instead we look at it’s energy or signal energy
• But this can be infinite!
• Not to be confused with mechanical Energy -- Not in Joules
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Signal Energy
ELEC 3004: Systems
• Generalize this to a finite measure via a RMS (root means
square measurand)
• Not to be confused with mechanical Power -- Not in Watts
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Signal Power:
Finite Energy
Finite Power
∞ Energy
Finite Power
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• Classifications – A “pair-wise” way of characterizing the signal by putting it into
a “descriptive bin”
– Ex: How to describe a person – well we could measure them
(weight, height, power, etc.) or we could sort by category
(“North/South-side,” Australian, etc.) Neither is perfect
• Some common classifications: 1. Continuous-time and discrete-time signals
2. Analog and digital signals
3. Periodic and aperiodic signals
4. Real and complex signals
5. Deterministic and probabilistic signals
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Signal Classifications
ELEC 3004: Systems
• The independent variable (x-axis) – in this case t – is
continuous (which may be the ℝeals, but can be others)
• This does not dictate the form of u(t) -- it may be either
continuous or discrete.
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Continuous-Time
u(t)
t
t R
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• The independent variable (x-axis) – in this case t – is
continuous (which may be the ℝeals, but can be others)
• This does not dictate the form of u(t) -- it may be either
continuous or discrete.
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Continuous Time
u(t)
t
t R
Continuous
ELEC 3004: Systems
• The independent variable is only set for fixed, discrete values
• Ex: t ∈ Integers
• Written in [Square Brackets]
• u[tn] = u[n]
• tn={t0, t1, t2, …, tn}
• Relationship to continuous time: u[n]=u(nTs)
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Discrete Time
u[t ]n
t
t Z (Integers)
Ts = Sample Period
= Time between samples
Discrete
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• Continuous and Discrete on the y-axis
• u(t) = Continuous Analog
• u(t) = Discrete Digital – Number of levels can be many
– Simplest case is for two Binary digit (or bit) signal
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Analog and Digital Signals
u(t)
t
Discrete
digital
levels
ELEC 3004: Systems
• (a) analog, continuous time | (b) digital, continuous time
• (c) analog, discrete time and (d) digital, discrete time
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Analog and Digital Signals
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• A signal x(t) is periodic if for some positive constant T0
• For periodic signals x(t) of period T0 : – the area under x(t) over any interval of duration T0 is the same
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Periodic and Aperiodic Signals
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Real and Complex Signals
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• Deterministic – are those whose description is
known completely
• Random – Those signals whose descriptions is unknown incompletely via
statistical or probabilistic descriptions.
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Deterministic and Random
0 1 2 3 4 5 6 7-1
-0.5
0
0.5
1
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Ergodicity
1. Over time: multiple readings of a quantity over time
• “stationary” or “ergodic” system
• Sometimes called “integrating”
2. Over space: single measurement (summed) from multiple sensors each distributed in space
3. Same Measurand: multiple measurements
take of the same observable quantity by multiple, related instruments e.g., measure position & velocity simultaneously Basic “sensor fusion”
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Classifying Signals: Even and Odd
• Even
– Area:
Every signal can be expressed as a sum of even and odd components
• Odd
– Area:
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Symmetry
Line
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Signal Models
A “model” of signals
• Key functions include:
– Step
– Impulse
– Exponential functions play
• Basis for representing other
signals,
• Simplify many aspects of
the signals and systems.
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-4 -2 0 2 4
-0.2
0
0.2
0.4
0.6
0.8
1
x
sin( x)/( x)
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Signal Models
1] Unit Step
• if A=1, t0=0
– Heaviside function
2] Unit Impulse
• Can be approximated as:
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• Multiply F(t) by δ(t)
• Is about t=0: – F(t) is continuous at t=0
– δ only has value at t=0
– ℶ=F(0)=F(t=0)
• Sampling Property of the Impulse:
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Signal Models: Multiplication by Impulse
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3] Sinusoidal & Exponential Signals
• A, ω : Amplitude and Frequency (ω=2πf)
• ϕ: phase angle
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Signal Models
ELEC 3004: Systems
Systems
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1. Linear and nonlinear systems
2. Constant-parameter and time-varying-parameter systems
3. Instantaneous (memoryless) and dynamic (with memory)
systems
4. Causal and noncausal systems
5. Continuous-time and discrete-time systems
6. Analog and digital systems
7. Invertible and noninvertible systems
8. Stable and unstable systems
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Further Classifications of Systems
ELEC 3004: Systems
• Modelling order depends on what you are trying to achieve
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Modelling order
http://ssd.jpl.nasa.gov/
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• Linear Circuit Theorems, Operational Amplifiers
• Operational Amplifiers
• Capacitors and Inductors, RL and RC Circuits
• AC Steady State Analysis
• AC Power, Frequency Response
• Laplace Transform
• Reduction of Multiple Sub-Systems
• Fourier Series and Transform
• Filter Circuits
Modelling Tools!
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Modelling Ties Back with ELEC 2004
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Example: RC Circuits
R
C
y(t)=
ΔV(t)
AC
f(t)=i(t)
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• Passive, First-Order Resistor-Capacitor Design:
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First Order RC Filter
• 3dB (½ Signal Power):
• Magnitude:
• Phase:
OutInR
C
(Low-pass configuration)
ELEC 3004: Systems
• KCL:
• KVL:
• Combining:
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Example of 2nd Order: RLC Circuits
L
R2
C
R1
+
−v S( t )
i ( t )
•
•
v C( t )
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• 2nd Order System Sallen–Key Low-Pass Topology:
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2nd Order Active RC Filter (Sallen–Key)
R2 C1 +
–C2
VinVout
• KCL:
• Combined with Op-Amp Law:
• Solving for Gives a 2nd order System:
Build this for
Real in
ELEC 4403
ELEC 3004: Systems
Another 2nd Order System: Accelerometer or Mass Spring Damper (MSD)
• General accelerometer: – Linear spring (k) (0th order w/r/t o)
– Viscous damper (b) (1st order)
– Proof mass (m) (2nd order)
Electrical system analogy: – resistor (R) : damper (b)
– inductance (L) : spring (k)
– capacitance (C) : mass (m)
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Measuring Acceleration: Sense a by measuring spring motion Z
• Start with Newton’s 2nd Law:
• Substitute:
•
• Solve ODE:
The “displacement”
measured by the unit
(the motion of m relative the
accelerometer frame)
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Measuring Acceleration [2] • Substitute candidate solutions:
• Define Natural Frequency (ω0)
& Simplify for Z0 (the spring displacement “magnitude”):
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Acceleration: 2nd Order System
• Plot for a “unit” mass, etc….
• For ωω0:
it’s a Seismometer Accelerometer Seismometer
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Equivalence Across Domains
Source: Dorf & Bishop, Modern Control Systems, 12th Ed., p. 73
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Source: Dorf & Bishop, Modern Control Systems, 12th Ed., p. 74
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• We’ll talk about System Models
• Review: – Phasers, complex numbers, polar to rectangular, and general
functional forms.
– Chapter 1 of Lathi
(particularly the first sections on signals & classification thereof)
• Register on Platypus
• Try the practise assignment (will be posted soon)
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Next Time…
ELEC 3004: Systems
